6. (i) Solve the congruence relation 3x + 5 = 2(3 – 5x) (mod 35) for r. - (ii) If a = b (mod n) and c =d (mod n), show that ac = bd (mod n). (iii) Z39, the set of all the reduced residue classes modulo 39.

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: (10) Find all solutions of the congruence 2 = 22 (mod 53). C

A: We shall be using primitive roots to solve this congruence.

Q: Let p be an odd prime. Prove that (a) If p = 1 (mod 5) or p = 4 (mod 5), then 5 is quadratic residue…

A: Given p is an odd prime. We have to prove (a). if p≡1(mod 5)  or  p≡4(mod 5), then 5 is a…

Q: (7) (a) Find the order of a modulo 7 for a = 1, 2, 3, 4, 5, 6.

A:

Q: If x = y (mod n) and u = v (mod n) then which of the following is false. (a) x+ u = y +v (mod n) (b)…

A: Given that x≡ymod n and u≡v mod n. By definition, x≡ymod n implies x-y is divisible by n. Then let,…

Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: For each statement, indicate whether it is true or false. Then prove or disprove it. (a) For all SC…

A: (a) FALSE.  Consider S ⊆ℤ, defined by  S = { 2+8k | k ∈ℤ } U { 2+12r | r∈ℤ }.  Then the given…

Q: Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer c…

A:

Q: If al = b (mod p), then prove that a = b (mod p).

A: (ii) given that                                                ap≡bp   (mod p) we have to prove…

Q: (i)Find the inverse of 5(mod9)     (  ii)Hence or otherwise solve simaltaneous equation…

A:

Q: ve squares modulo p) We can use quadratic reciprocity to answer tr question: "How many residue class…

A: Given:- ∑x=0p=1xp=0 and if that x does not equal to zero (mod p)…

Q: (6) Compute ind2 (19) modulo 101.

A: We recall the definition of a primitive root.

Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

A:

Q: (2). What is the binary representation of the following: (a) (64.37109375) 10.

A:

Q: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

A:

Q: For each statement below decide if it is TRUE or FALSE. • There are two integers n and m such that…

A: We will find out whether the given statements are true or false.

Q: 2 has order m modulo (2m - 1).

A: To prove that 2 has order m modulo , we need to show two things:For any positive integer

Q: SImalest posiuiy OI the n) is order of a modulo n. Find the additive orders of each of the following…

A:

Q: Make sure that all your answers are explained carefully. (a) Is 2 a quadratic residue modulo 13?…

A:

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

A:

Q: () 2p n! (a) (") = 2 (mod p), where k for 0 <k <n. k!(n – k)! (b) (Зр - 1)! 2р2 (mod p3)

A: As per our guidelines we are supposed to answer only fast question. Kindly repost other part in the…

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

A:

Q: Compute the following (over Z2[x])(a) x3(x2 + x + 1) (mod x3 − 1).(b) (x2 + 1)2 (mod x3 − 1)

A: Step 1:To compute these expressions over Z2[x], we need to perform polynomial arithmetic under…

Q: (a) Let p E N be a prime number, p # 5, and let nE N. Determine the number of solutions to the…

A: First note that this question is only solvable for certain primes and not for all primes. We…

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A:

Q: 2.3(i) Find the conjugates, moduli and arguments (to the nearest degree) of the following

A: 2.3(i)   (a). Given,                         3+5i.     To find the conjugate, modulus and argument…

Q: a) Find a0, a1, a2, a3 so that 1234 = a3 · 10^3 + a2 · 10^2 + a1 · 10^1 + a0 · 10^0 b) Prove that…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: . (a) How many quadratic residues are in the residue set Z19? (b) Determine if 4 and 9 are quadratic…

A: Introduction: Quadratic residue is related to congruent modulo theory. Since, it deals with…

Q: 13. Suppose that z1 V3 +i and z2 = 2/2+ 2v2i. 22 (a) Write z1 and z2 in mod-arg form. (b) Hence…

A:

Q: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

A:

Q: 9.Let be a generating function defined on [-3, 7) for the periodic function .  Write an expression…

A:

Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…

Q: Q-2: a) Four married couples have bought 8 seats in the same row for a concert. In how many…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Using appropriate methods, reduce (i) 419 mod 47, (ii) 141-10156 mod 47, (iii) 483 mod 47, (iv) (419…

A: We have to reduce the following congruent equation.

Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

A:

Q: Let a, b and c a=1b=0C=9 example with enrolment number 001054321, you would take a = 0, b = 5 and c…

A: The answer to the given question is provided below with respective explanation.

Q: (c) Find all integers of order 6 mod 13 and all primitive roots mod 13. Show your working in both…

A: This is a question of Number Theory.

Q: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

A: I have proved it by using defination of congruent.

Q: 6. Let f(x) be a polynomial with integral coefficients (like f(x) = 3x^^5-x^2+7) and suppose there…

A: Given:- fx=3x5-x2+7 There are 2 solutions  x mod 5 such that fx is congruent to 3 mod 5 and there…

Q: Show that if p = 1 (mod 4), then x² = -1 (mod p) has a solution given by the least residue (mod p)…

A: Given: p≡1 mod 4We need to prove that x2≡-1 mod p has a solution given by the least residue mod p of…

Q: Find a number x between 0 and 90, such that x ≡ [4,8] mod [7,13].

A: